Purpose
and Goals: The purpose of this one credit Math
410 course is to introduce students to possible undergraduate research
projects in computational mathematics at William and Mary. The
format will consist mainly of weekly talks by faculty (approximately 30
minutes) followed by class discussions and/or exercises related to the
presented topics. The typical student in this course will be in
his or her sophomore or junior year and will have an interest in
pursuing a research project related to computational mathematics.
For many, this course can serve as a gateway to establishing a research
project through the CSUMS
program, with applications for this program due at the end of the
spring semester. If you have any questions about whether this
course could be appropriate for you, please contact me
or one of the faculty members listed on the CSUMS
website. (Students who have previously taken ``Math
410: Topics in Computational Mathematics'' are welcome to enroll again
this semester. Students may petition the Chair of the
Mathematics Department to have three 1 credit Math 410 courses count as
one 400 level course for the mathematics major.)

Course
Grade: The course grade will be based on attendance
and participation. Students may miss 1 of the 13 talks without
penalty. Students may earn extra credit for attending CSUMS/Math
Department colloquia and other appropriate talks listed
below.
Attendence of classes and colloquium talks will be recorded by the
organizers, and participation are shown from discussion after the talk
and "discussion board" on Blackboard website. More specifically, with
total of 100 points, the attendance of each talk is 4 points,
discussion in class is 2 points, and discussion on Blackboard is 2
points. Attendance of each Math colloquium is 2 extra points.

Abstract: A network of oscillators is an
effective model in the study of neural synchronization.
In this talk,
we initially explore the effect of correlations between the in- and
out-degrees
(i.e. node-degree correlations) of random directed networks
on the synchronization of identical
pulse-coupled oscillators. We
demonstrate a variety of results through numerical experiments,
for
example networks with negative node-degree correlation are less likely
to achieve global
synchrony and synchronize more slowly than networks
with positive node-degree correlation.
We then show how this effect of
node-degree correlation on synchronization of pulse-coupled
oscillators
is consistent with aspects of network topology (e.g., Laplacian
eigenvalues, clustering
coefficient) that have been shown to affect
synchronization in other contexts. Finally, we end
with a more
in-depth look into the global dynamics on all strongly connected 3-node
networks.

Abstract: Given a graph $G$, an
identifying code $\code\subseteq V(G)$ is a
vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$,
the sets $N[v_1]\cap\code$ and $N[v_2]\cap\code$ are distinct and
nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We
study the case when $G$ is the infinite hexagonal grid $H$. Cohen
et.al. constructed two identifying codes for $H$ with density $3/7$
and proved that any identifying code for $H$ must have density at
least $16/39\approx0.410256$. Both their upper and lower bounds were
best known until now. Here we prove a lower bound of
$12/29\approx0.413793$.

Week 12 (4/7)

Andrew Wilcox

Week 13 (4/14)

Distance geometry and
biomolecular structure estimation

Michael
Lewis

We discuss some applications of
distance geometry in the determination of the
structure of proteins and DNA. Distance geometry refers to the
characterization
of a set of points using information on the distance between the points.

Distance geometry can be used to estimate protein structure from lower
and
upper bounds on interatomic distances determined by nuclear Overhauser
effect
spectroscopy (NOESY). NOESY data is sparse, however, and more
realistic
protein structure determination requires the minimization of an
empirical
energy function. As we discuss, parameterization of the energy
function in
terms of interatomic distances leads to a more tractable optimization
problem.

Another application of distance geometry is the Partial Digest Problem.
In this problem an enzyme is used to cut a batch of DNA strands at
locations
known as restriction sites (though not every site need be cut).
The resulting
mix contains fragments whose lengths correspond to a subset of the
distances
between restriction sites. The question is then one of
reconstructing the
original sequence from this distance data.

In the talk we will discuss these and other applications, and how they
can be
attacked using a combination of distance geometry and optimization.
The
approaches we will discuss result are large-scale, nonconvex
optimization
problems that involve functions of the eigenvalues of matrices.